Contents Ltest CBSE Smple Pper -7 Solution to Ltest CBSE Smple Pper -7 Prctice Pper Solution to Prctice Pper 8 Prctice Pper 9 Solution to Prctice Pper Unsolved Prctice Pper 7 Unsolved Prctice Pper 7 Unsolved Prctice Pper 79 Unsolved Prctice Pper 8 Answers to Unsolved Prctice Ppers 8 Mock Emintion Pper 7 9
Complete Solutions of Unsolved Prctice Ppers nd Mock Emintion Pper For Android Users Downlod the ndroid pp Gupt Bnsl from Google Pl Store or wwwguptnslcom Scn the QR codes given elow Unsolved Pper Unsolved Pper Unsolved Pper Unsolved Pper Mock Emintion Pper For Other Users Visit the wesite wwwguptnslcom or wwwsultnchndeookscom/wsmthemtics-xii
Complete Solutions of Unsolved Prctice Ppers nd Mock Emintion Pper For Android Users Downlod the ndroid pp Gupt Bnsl from Google Pl Store or wwwguptnslcom Scn the QR codes given elow Unsolved Pper Unsolved Pper Unsolved Pper Unsolved Pper Mock Emintion Pper For Other Users Visit the wesite wwwguptnslcom or wwwsultnchndeookscom/wsmthemtics-xii
Ltest CBSE Smple Pper -7 Time: Hours M Mrks: Generl Instructions (i) All questions re compulsor (ii) Plese check tht this question pper consists of 9 questions (iii) Questions to in Section A re Ver Short Answer Tpe Questions crring mrk ech (iv) Questions to in Section B re Short Answer I Tpe Questions crring mrks ech (v) Questions to in Section C re Long Answer I Tpe Questions crring mrks ech (vi) Questions to 9 in Section D re Long Answer II Tpe Questions crring mrks ech (vii) Plese write down the seril numer of the question efore ttempting it Section A Question numers to crr mrk ech Stte the reson wh the reltion R {(, ) : } on the set R of rel numers is not refleive If A is squre mtri of order nd A k A, then find the vlue of k If ~ nd ~ re two non-zero vectors such tht ~ ~ ~ ~, then find the ngle etween ~ nd ~ If is inr opertion on the set R of rel numers defined, then find the identit element for the inr opertion Section B Question numers to crr mrks ech Simplif: cot, for < Prove tht the digonl elements of skew smmetric mtri re ll zeros 7 If tn, < <, then prove tht 9 8 If chnges from to, then find the pproimte chnge in loge e 9 Find Otin the differentil eqution of the fmil of circles pssing through the points (, ) nd (, ) If ~ ~, ~ ~ nd ~, then find ~ If P (A), P (B), P (A B), then find P (A B)
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Section C Question numers to crr mrks ech, then using A, solve the following sstem of equtions: If A, if < t Discuss the differentiilit of the function f () if For wht vlue of k is the following function continuous t? sin cos if f () k if d If sin pt, cos pt, then show tht ( ) Find the eqution of the norml to the curve,which psses through the point (, ) i Seprte the intervl, into suintervls in which the function f () sin cos is strictl incresing or strictl decresing h 7 A mgzine seller hs suscriers nd collects nnul suscription chrges of per suscrier She proposes to increse the nnul suscription chrges nd it is elieved tht for ever increse of, one suscrier will discontinue Wht increse will ring mimum income to her? Mke pproprite ssumptions in order to ppl derivtives to rech the solution Write one importnt role of mgzines in our lives sin 8 Find (cos )(cos ) 9 Find the generl solution of the differentil eqution ( tn )( ) Solve the following differentil eqution: e/ e/ Prove tht: ~ {(~ ~c) (~ ~ ~c)} [ ~ ~ ~c ] Find the vlues of so tht the following lines re skew: z, z A g contins green nd white lls Two lls re drwn one one without replcement If the second ll drwn is white, wht is the proilit tht the first ll drwn is lso white?
Ltest CBSE Smple Pper Ji B Ki Two crds re drwn successivel with replcement from well shuffled pck of crds Find the proilit distriution of the numer of dimond crds drwn Also, find the men nd the vrince of the distriution Section D Question numers to 9 crr mrks ech Let f : [, ) R e function defined f () 9 Prove tht f is not invertile Modif, onl the codomin of f to mke f invertile nd then find its inverse Let e inr opertion defined on Q Q (, ) (c, d) (c, d), where Q is the set of rtionl numers Determine, whether is commuttive nd ssocitive Find the identit element for nd the invertile elements of Q Q ( ) c c c ( c) ( c) Using properties of determinnts, prove tht (c ) p q pα q If p, q nd q r qα r, then using properties of determinnts, prove pα q qα r tht t lest one of the following sttements is true: () p, q, r re in G P, () α is root of the eqution p q r Using integrtion, find the re of the region ounded the curves nd / sin cos 7 Evlute: sin cos Evlute: e s the limit of sum 8 Find the eqution of the plne through the point (,, ) nd perpendiculr to the line of intersection of the plnes z nd z Find the point of intersection of the line ~r i j k λ(i j 9k ) nd the plne otined ove 9 In mid-d mel progrmme, n NGO wnts to provide vitmin rich diet to the students of n MCD school The dieticin of the NGO wishes to mi two tpes of food in such w tht vitmin contents of the miture contins t lest 8 units of vitmin A nd units of vitmin C Food contins units per kg of vitmin A nd unit per kg of vitmin C Food contins unit per kg of vitmin A nd units per kg of vitmin C It costs per kg to purchse Food nd 7 per kg to purchse Food Formulte the prolem s LPP nd solve it grphicll for the minimum cost of such miture? ooo
Solution to Ltest CBSE Smple Pper -7 Given R Set of ll rel numers nd R {(, ) : ;, R} Refleive: Since,, is not true, for some R For, we hve (, ) / R So, R is not refleive Given tht A is mtri of order nd A k A We hve, A k A A k A 8 A k A k 8 [ αa αn A, where A is n n mtri] [ A ] Given two non-zero vectors ~ nd ~ such tht ~ ~ ~ ~ Let θ e the ngle etween ~ nd ~ Now, ~ ~ ~ ~ ~ ~ sin θ ~ ~ cos θ sin θ cos θ tn θ [ ~, ~ ] θ Hence, ngle etween ~ nd ~ is Given tht is inr opertion on R defined, for ll, R Eistence of Identit: Let e R e the identit element, then for ever R e nd e e nd e e nd e So, e R Hence, hs identit e on R Let cot, where < Putting sec θ, we get cot cot cot (cot θ) θ sec tn θ sec θ Hence, sec is the required simplest form Let A [ij ] e skew smmetric mtri AT A
Solution to Ltest CBSE Smple Pper T [ij ] [ij ] [ji ] [ij ] ji ij ii ii for ll possile vlues of i, j for ll possile vlues of i ii for ll possile vlues of i ii for ll possile vlues of i Hence, ll the digonl elements of A re zero 7 Let tn tn () tn () tn () () Differentiting oth sides wrt, we get d d () () d d tn () tn () () () 9 8 Let loge Choose nd Then, (i) Then, Now, ()() (ii) Hence, required pproimte chnge in loge is 9 e ( ) ( ) e ( ) ( ) {z} e e ( ) II {z } I ( ) ( ) e e e ( ) ( ) e C e A circle which psses through the points (, ) nd (, ) hs its centre on -is Let (, ) e the centre of the circle Then, rdius of circle Distnce etween (, ) nd (, ) Ji B Ki
Supplement to An Aid to Mthemtics Clss XII Ji B Ki So, the eqution of the fmil of circles which pss through the points (, ) nd (, ) is ( ), where is n ritrr constnt Differentiting oth sides wrt, we get, which is the required differentil eqution Given ~ ~, ~ ~ nd ~ Now, ~ ~ (~ ~) (~ ~) (~ ~) Now, ~ ~ (~ ~) (~ ~) (~ ~) ~ (~ ~) ~ (~ ~) ~ ~ ~ ~ ~ ~ ~ ~ ~ (~ ~) ~ (~ ~) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ () () ~ ~ ~ 8 ~ ~ ~ ~ ~ ~ () ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ () () ~ ~ ~ 8 ~ ~ ~ ~ ~ ~ On dding () nd (), we get ~ ~ ~ Given P (A), P (B) nd P (A B) Then, P (A B) P (A B) P (A B) P (A B) P (B) P (B) P (B) [P (A) P (B) P (A B)] P (B) 7 ()
Solution to Ltest CBSE Smple Pper Given A Then, A So, A is non-singulr nd hence invertile Let Aij denote cofctor of ij in A [ij ], then A ( ) (), A ( ) ( ), A ( ) (), A A dj A A A (dj A) nd A A Now, given sstem of equtions is A ( ) () Ji B Ki7 which cn e written in mtri form s AX B, ie, where A,X,B Since, A Given sstem of equtions is consistent nd hs unique solution given X A B X A B, Hence, the required solution is, if < Given f () if Differentiilit t f h f Rf lim h h f h f Lf lim h h
Supplement to An Aid to Mthemtics Clss XII h h lim lim h h h h 8Ji B Ki h lim lim [ ] h h h Since, Rf h lim lim [] h h h Lf Hence, f is not differentile t sin cos Given f () k Since, f is continuous t lim ( /) [f ()] if if [f ()] f ( /) lim f lim [f ()] ( /) sin cos k lim ( /) ) ( sin cos lim ( /) sin sin cos lim ( /) n o sin lim ( /) n o sin h lim h h sin h lim h h () cos h B putting i h
Solution to Ltest CBSE Smple Pper Given sin pt cos pt nd Differentiting oth sides wrt t, we get p cos pt dt Ji B Ki9 () Differentiting oth sides wrt t, we get p sin pt dt p sin pt dt Then, tn pt p cos pt dt Differentiting oth sides wrt, we get d d [tn pt] dt (p sec pt) (p cos pt) cos pt cos pt sin pt) cos pt ( sin pt)( cos pt) d ( ) ( ( ) ( ) [Using ()] d d Given curve is Let the point of contct of required norml with the given curve e (α, β) β α () Differentiting oth sides of () wrt, we get Now, eqution of norml t (α, β) is () α (α,β) β ( α) (α,β)
Supplement to An Aid to Mthemtics Clss XII Ji B Ki β ( α) α Given tht norml line psses through the point (, ) β ( α) α α ( α) α ie, () [Using ()] α α α α α / / / / Putting α / nd β / in (), we get Eqution of norml t /, / is Putting α / in (), we get β / / ( / ) ie, / / ie, / / i Given f () sin cos on, h Then, f () sin cos cos sin sin cos (cos sin ) (sin )(cos ) sin Now, f () sin sin,,, (A) h i,, (B) / / Test Vlue Sign of f () (A) f ( ) (B) f () Hence, f () is strictl incresing on, nd strictl decresing on, Su-intervl,, Test Point Conclusion f () < f () >
Solution to Ltest CBSE Smple Pper Ji B Ki 7 Suppose tht the mgzine seller increses the nnul suscription per suscrier Then, the numer of suscriers who will discontinue the services re Now, totl revenue of the compn fter the increment is given R Numer of suscriers left New suscription chrges per suscrier ( )( ) dr dr Now, Then, d R Also, d R < So, second derivtive test, R is mimum when Hence, required increse in the suscription chrges per suscrier is Mgzine suppl us with vriet of news nd contriute in the development of our knowledge sin 8 Let I (cos )(cos ) Put cos sin I ( )( ) Putting t in the frction Put (), we get ( ) ( ) ( ) ( ) (t )(t ) A B (t )(t ) t t A(t ) B(t ) At A Bt B () On compring coefficients of t nd constnt terms on oth sides, we get A B, A B On solving ove equtions, we get A, B
Supplement to An Aid to Mthemtics Clss XII Ji B Ki Putting vlues of A nd B in (), we get (t )(t ) (t ) (t ) ( ) ( ) ( ) ( ) From () nd (), we get I tn tn C cos tn (cos ) tn C () 9 Given differentil eqution is ( tn )( ) ( tn ) ( tn ) tn, tn P Q with P nd Q which is of the form tn Now, integrting fctor, IF e R P e R tn e R cos cos sin e R (cos sin )(cos sin ) cos sin cos sin e ( sin cos ) R elog (sin cos ) e (sin cos ) Generl solution of given differentil eqution is (IF) Q(IF) C e (sin cos ) e (sin cos ) C e (sin cos ) {z} e sin e cos C {z} II e (sin cos ) sin I e e (sin cos ) e sin e (sin cos ) e sin C, which is the required solution d {sin } [(cos )(e )] e e cos C e cos C
Solution to Ltest CBSE Smple Pper Ji B Ki Given differentil eqution is ( e/ ) e/ ( e Putting v nd / ) e / e e/ / () dv v in (), we get ev (v ) dv v ev ev (v ) dv v ev dv ev v ev v e dv v ve Integrting oth sides, we get ev dv v ev log v ev log C log e/ log C, which is the required solution LHS ~ {(~ ~c) (~ ~ ~c)} ~ {(~ ~) (~ ~) (~ ~c) (~c ~) (~c ~) (~c ~c)} ~ {(~ ~) (~ ~c) (~c ~) (~c ~)} ~ ~ ~ ~c ~c ~ ~ (~ ~) ~ (~ ~c) ~ (~c ~) ~ (~c ~) ~ (~ ~c) ~ (~c ~) ~ (~ ~c) ~ (~ ~c) ~ (~ ~c) [ ~ ~ ~c ] RHS The Crtesin equtions of given lines re z nd z [ ~c ~ ~ ~c ]
Supplement to An Aid to Mthemtics Clss XII Ji B Ki The vector equtions of given lines re ~r (i j k ) λ(i j k ) nd ~r (i j k ) µ(i j k ) ~ λ~ ~ µ~ ~ i j k, ~ i j k ~ i j k, where ~ i j k, Here, ~ ~ Given lines re either intersecting or skew lines It is given tht the given lines re skew lines So, the given lines re non-intersecting Eqution of first line is z z Let α Then, α, α, z α Generl point on first line is (α, α, α ) () Eqution of second line is z Let z β Then, β, β, z β Generl point on second line is (β, β, β) () On equting the coordintes of () nd (), we get α β α β () α β α β () α β α β () From () nd (), we get α, β Putting α, β in (), we get So, the given lines intersect if nd the given lines do not intersect if Hence, the given lines re skew if Given Bg : green nd white lls Let B, B nd E e the events defined s follows: B : First ll is green, B : First ll is white, P (E B ) P (B ), P (E B ) B Bes Theorem, we hve Then, P (B ) nd E : Second ll is white, 9 9 P (B )P (E B ) 9 P (B E) P (B )P (E B ) P (B )P (E B ) 9 9 9 Hence, required proilit 9
Solution to Ltest CBSE Smple Pper Ji B Ki Totl numer of crds Numer of crds drwn (with replcement) Suppose tht numer of dimond crds is considered s success Let X e rndom vrile defined s the numer of successes Then, X cn ttin the vlues,, Now, P (X ) P (oth re non-dimond crds) 9 9 9 P (X ) P (one is dimond crd nd one is non-dimond crd) [Fi Cse] [Unfi Cse]! P (first is dimond crd nd net is non-dimond crd)!! 9!!! P (X ) P (oth re dimond crds) So, proilit distriution of X is given X 9 P (X) [Fi Cse] Clcultion of Men nd Vrince X P (X) 9 P Hence, men X X X X P (X) X P (X) X P (X) vrince vr(x) X P (X) 8 P X P (X), X P (X) hx i X P (X) 8 8 Given f : [, ) R defined f () 9 (i) One-one: Let, [, ) e n two elements f ( ) f ( ) Then, 8 9 9 9 9 9( ) ( ) ( )[9( ) ]
Ji B Ki Supplement to An Aid to Mthemtics Clss XII [ 9( ) ] So, f is one-one (ii) Onto: Let R e n element Then, f () 9 9 p (9)( ) 8 p ± ( ) 8 p ± ( ) 8 ± 8 ± ± For 7 R, we hve / [, ) So, f is not onto Hence, f is not ijection nd hence not invertile We know tht f is onto if the co-domin of f equls its rnge Now, f () 9 ( ) We hve, < < < ( ) < ( ) < f () < So, Rnge (f ) [, ) So, the modified function is f : [, ) [, ) We know tht ever function is onto up to its rnge So, f is onto Thus, f is ijection nd hence invertile [, ) For ever [, ), we hve Since, f : [, ) [, ) eists nd from (), we hve f () Hence, inverse of f is given f () () [ f () f ()]
Solution to Ltest CBSE Smple Pper Ji B Ki7 Given inr opertion on Q Q, defined (, ) (c, d) (c, d) Commuttivit: Let (, ), (c, d) Q Q e n two elements Then, (, ) (c, d) (c, d) nd (c, d) (, ) (c, d c) For (, ) (, ), (c, d) (, ), we hve (, ) (c, d) (, ) nd (c, d) (, ) (, ) (, ) (c, d) (c, d) (, ) Hence, is not commuttive on Q Q Associtivit: Let (, ), (c, d), (e, f ) Q Q e n three elements Then, ((, ) (c, d)) (e, f ) (c, d) (e, f ) (ce, d cf ) nd (, ) ((c, d) (e, f )) (, ) (ce, d cf ) (ce, d cf ) ((, ) (c, d)) (e, f ) (, ) ((c, d) (e, f )) Hence, is ssocitive on Q Q Eistence of Identit: Let (e, e ) Q Q e the identit element, then for ever (, ) Q Q (, ) (e, e ) (, ) nd (e, e ) (, ) (, ) (e, e ) (, ) nd (e, e e ) (, ) e, e nd e, e e e, e nd e, e (e, e ) (, ) Q Q Hence, hs identit (e, e ) (, ) on Q Q Eistence of Inverse: Let (, ) Q Q e n invertile element nd let (c, d) Q Q e its inverse, then (, ) (c, d) (e, e ) nd (c, d) (, ) (e, e ) (c, d) (, ) c, d nd nd (c, d c) (, ) c, d c,d nd c c,d nd c Since, (c, d) Q Q for ever (, ) Q Q with c, d c,d So, (, ) with re the invertile elements of in Q Q nd their inverses re LHS ( ) c c c ( c) (c ),
Supplement to An Aid to Mthemtics Clss XII 8Ji B Ki c c ( ) c c c ( c) (c ) c c( ) c c c c ( ) c c c ( c) ( c) ( ) c c c c c (c ) (c ) ( c) ( c)( c) c On tking common c from R, from R nd from R ( c) c c c c c c On ppling R R R R R R c c c c c On tking common ( c) from R nd R ( c) c ( c) c ( c) c [ c {( )( c) c}] c c c c [c { c}] ( c) RHS Let p q pα q q r qα r pα q qα r On ppling C C C C On ppling R R R c R R R h i On epnding long C ( c )( c ) ( c )( c ) ( c )( c ) (c ) c c c c c (c ) ( c) c (c ) (c ) (c ) ( c) c On multipling C c, C nd C
Solution to Ltest CBSE Smple Pper p q pα q q r qα r Ji B Ki9 pα qα r p ( pα qα r) q pα q q r qα r On ppling C C αc C On tking common ( pα qα r) from C h ( pα qα r)[ {pr q }] On epnding long C i (q pr)(pα qα r) Given tht (q pr)(pα qα r) either q pr or pα qα r Hence, either p, q, r re in G P or α is root of the eqution p q r ( if Given curves re nd if Given curves re (i) (ii), if (iii), if To find points of intersection From (i) nd (ii), we hve p From (i) nd (iii), we hve p ( )( ) ( )( ) [ ] Curves (i) nd (ii) intersect ech other t (, ) [ ] Curves (i) nd (iii) intersect ech other t (, ) To plot the required region (if ) (if ) The required region is shown shded in the grph
Supplement to An Aid to Mthemtics Clss XII Ji B Ki R Scle Along -is: cm unit R Along -is: cm unit To find the re of required region A Are of region R Are of region R hp hp i i ( ) ( ) p p sin sin sin sin sin sin sin sin sin sin sin sin sin sq units Hence, required re / sin cos 7 Let () I sin cos / sin cos I sin cos / cos sin I () cos sin On dding () nd (), we get / cos sin / sin cos I sin cos cos sin / sin cos I sin cos
Solution to Ltest CBSE Smple Pper I / sin cos sin cos / tn sec tn Ji B Ki On dividing numertor nd denomintor cos tn Put If, then If, then tn sec i h tn I 8 8 8 Given integrl is e We know tht f () lim h [f () f ( h) f ( h) f ( (n )h)], h where n h Choose,, f () e, then e lim h [f () f ( h) f ( h) f ( (n )h)] h h h Now, f ( hp) ( hp) e(hp) hp ehp where n () () () Putting p,,,, (n ) successivel in (), we get f () f ( h) h eh f ( h) h eh f ( (n )h) (n )h e(n )h Adding ove n equtions, we get f () f ( h) f ( h) f ( (n )h) [ eh eh e(n )h ] h[ (n )] n eh (n )n h eh 8 e h h e h h 8 e h e h e8 ( h) h e h [Using ()]
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Putting this vlue in (), we get 8 e e lim h h ( h) h e h h 8 ( h) lim (e ) h h e 8 h e8 e ( h) 8 lim h h e 8 Let,, c e the direction rtios of norml to the required plne Given tht the required plne is perpendiculr to the line of intersection of the plnes z nd z Direction rtios of the line of intersection of given plnes re,, c Also, direction rtios of norml to the first plne re,, nd direction rtios of norml to the second plne re,, c nd c () () On solving () nd () cross multipliction, we get c c ie, k (s) 7 k, 7k, c k Also, the required plne psses through the point (,, ) Let,, z Hence, eqution of the required plne is ( ) ( ) c(z z ) ie, k( ) 7k( ) k(z ) ie, ( ) 7( ) (z ) ie, 7 z Now, given eqution of line is ~r i j k λ(i j 9k ) So, eqution of the given line in Crtesin form is z 9 z Let k 9 Then, k, k, z 9k From () nd (), we get (k ) 7(k ) ( 9k ) k Putting k in (), we get,, z 8 Hence, the point of intersection of the given line nd required plne is (,, 8) () ()
Solution to Ltest CBSE Smple Pper Ji B Ki 9 Let the quntit of Food kg nd the quntit of Food kg Let totl cost We cn represent the given LPP in the following tulr form: Food Food Requirement Cost ( ) 7 Minimise Vitmin A (units) At lest 8 Vitmin C (units) At lest Minimise 7 Hence, given LPP is, suject to the constrints: 8,,, We consider the following equtions: 8 8, The fesile region of LPP is unounded, s shown shded in the grph Scle Along -is: cm unit P 8 Along -is: cm unit 7 Q O 8 7 8 9 R 7 < 8
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Corner Points P (, 8) Q(, ) R(, ) Vlue of ( 7) 8 Since, the fesile region is unounded nd 8 is the minimum vlue of t corner points So, we consider the open hlf plne 7 < 8, which hs no point in common with the fesile region 8 is the minimum vlue of in the fesile region t, Hence, quntit of Food kg, quntit of Food kg nd minimum cost 8 ooo
Prctice Pper Time: Hours M Mrks: Generl Instructions (i) All questions re compulsor (ii) Plese check tht this question pper consists of 9 questions (iii) Questions to in Section A re Ver Short Answer Tpe Questions crring mrk ech (iv) Questions to in Section B re Short Answer I Tpe Questions crring mrks ech (v) Questions to in Section C re Long Answer I Tpe Questions crring mrks ech (vi) Questions to 9 in Section D re Long Answer II Tpe Questions crring mrks ech (vii) Plese write down the seril numer of the question efore ttempting it Section A Question numers to crr mrk ech Evlute: tn sec ( ) If A nd B re squre mtrices of order nd A, B, then find the vlue of AB Find λ, if the vectors λi j k, i λj k nd j j λk re coplnr Let e inr opertion, on the set of ll non-zero rel numers, given for ll, R {} Find the vlue of, given tht ( ) Section B Question numers to crr mrks ech If the curve 7 nd cut orthogonll t (, ), then find the vlue of A g contins red mrles nd lck mrles Three mrles re drwn one one without replcement Wht is the proilit tht t lest one of the three mrles drwn e lck, if the first mrle is red? 7 Prove tht: cot 7 cot 8 cot 8 cot / tn 8 Evlute: m tn 9 Prove tht [ ~ ~ ~ ~c ~c ~ ] [ ~ ~ ~c ] Show tht Find the intervls on which the function f () tn (sin cos ) on, is (I) strictl incresing or strictl decresing (II) incresing or decresing sin Evlute: cos
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Section C Question numers to crr mrks ech Find the re of the qudrilterl ABCD, where A(,, ), B(,, ), C(,, ) nd D(,, ) Using vectors, find the vlue of k such tht the points (k,, ), (,, ) nd (,, ) re colliner r Evlute: sin mn If m n ( ), then prove tht Prove, using properties of determinnts: c c c c c c 7 Solve: ( )( ) 8 A letter is known to hve come either from TATANAGAR or CALCUTTA On the envelope just two consecutive letters TA re visile Wht is the proilit tht the letter hs come from TATANAGAR? Suppose tht % of the people with lood group O re left-hnded nd % of those with other lood groups re left-hnded % of the people hve lood group O If left-hnded person is selected t rndom, wht is the proilit tht he/she will hve lood group O? 9 Find the ngle etween the following pir of lines: z 8 z nd 7 nd check whether the lines re prllel or perpendiculr A sphericl ll of slt is dissolving in wter in such mnner tht the rte of decrese of the volume t n instnt is proportionl to the surfce Prove tht the rdius is decresing t constnt rte A ldder, m long, stnding on horizontl floor, lens ginst verticl wll If the top of the ldder slides downwrds t the rte of cm/sec, then find the rte t which the ngle etween the floor nd the ldder is decresing when lower end of ldder is m from the wll In hurdle rce, pler hs to cross hurdles The proilit tht he will cler ech hurdle is Wht is the proilit tht he will knock down fewer thn hurdles? Wht life skills should the pler develop to improve his performnce? Evlute: cos sin if Find the vlue of k so tht the function f () is continuous t k if
Prctice Pper Ji B Ki7 Section D Question numers to 9 crr mrks ech If A, find A Using A, solve the sstem of liner equtions:, z 8, z 7 Find the re of the region ounded the prol p nd p For rel numers nd, define R if nd onl if is n irrtionl numer Determine whether the reltion R is refleive, smmetric nd trnsitive Solve: sin () sin 7 If product of distnces of the point (,, ) from origin nd plne ~r (i j k ) p e 8, then find the vlue of p Find the length nd foot of perpendiculr from the point P (7,, ) to the plne z Also, find the imge of point P in the plne 8 An Apche helicopter of enem is fling long the curve given 7 A soldier, plced t (, 7), wnts to shoot down the helicopter when it is nerest to him Find the nerest distnce 9 A mn rides his motorccle t the speed of km/hour He hs to spend per km on petrol If he rides it t fster speed of 8 km/hour, the petrol cost increses to per km He hs t most to spend on petrol nd one hour s time Determine the mimum distnce tht the mn cn trvel Epress it s LPP nd then solve it ooo Find the re of the gretest rectngle tht cn e inscried in n ellipse
Solution to Prctice Pper tn sec ( ) tn sec () sec sec tn tn D n oe, ; [, ] Given tht A nd B re squre mtrices of order such tht A nd B Then, AB AB [ ka k n A, where A is n n mtri] A B (7)()() Let ~ λi j k, ~ i λj k nd ~c j j λk Since, the given vectors re coplnr Then, ~ (~ ~c) λ λ λ λ(λ ) (λ ) ( λ) λ λ λ λ λ λ (λ )(λ λ ) Given on R {}, defined Given λ, ± ( ) Given curves re 7 nd Also, given tht the point of intersection of the given curves is (, ) () ()
Solution to Prctice Pper Ji B Ki9 Differentiting oth sides of () nd () wrt, we get m (s) (,) Since, given two curves re orthogonl m (s) (,) m m () Numer of red mrles Numer of lck mrles Totl numer of mrles 8 Numer of mrles drwn (without replcement) Let E nd F e the events defined s follows: E : At lest one of the mrles is lck nd F : First mrle is red Then, E F : At lest one of the mrles is lck nd first mrle is red Also, P (F ), 8 P (E F ) P (RRB) P (RBR) P (RBB) 8 7 8 7 8 7 P (E F ) Hence, required proilit P (E F ) P (F ) 7 8 7 LHS cot 7 cot 8 cot 8 tn tn 7 8 8 tn 7 8 tn 8 7 8 tn tn 8 tn 8 8 tn cot RHS tn < 7 8 < 8
Ji B Ki / 8 Let I Supplement to An Aid to Mthemtics Clss XII / tn sin cos m tn cos m sin cos m sin Put ( cos sin m sin cos ) If, then If, then m sin cos (m ) sin cos I (m ) m (m ) m [log ] log m log (m ) (m ) [ log m ] (m ) log m m 9 LHS [ ~ ~ ~ ~c ~c ~ ] (~ ~) {(~ ~c) (~c ~)} (~ ~) {(~ ~c) (~ ~) (~c ~c) (~c ~)} (~ ~) {(~ ~c) (~ ~) (~c ~)} [ ~c ~c ~ ] ~ (~ ~c) ~ (~ ~) ~ (~c ~) ~ (~ ~c) ~ (~ ~) ~ (~c ~) ~ (~ ~c) ~ (~ ~c) ~ (~ ~c) [ ~ ~ ~c ] RHS LHS On tking common from C nd from C in first determinnt; on tking common from C nd from C in second determinnt
Solution to Prctice Pper Ji B Ki C nd C re identicl in first determinnt; () () C nd C re identicl in second determinnt RHS Given f () tn (sin cos ) on, Then, f () (cos sin ) (sin cos ) Now, f () (cos sin ) (sin cos ) cos sin tn, which is not true for n, (A) / Su-intervl (A), Test Point Test Vlue f ( ) ( ) Sign of f () Conclusion () f () > (I) f () is strictl incresing on, (II) f () is incresing on, sin Let I cos sin cos cos sec tn {z} {z } I II d sec () sec tn tn () tn tn tn tn tn tn C
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Given qudrilterl ABCD with vertices A(,, ), B(,, ), C(,, ) nd D(,, ) D Then, AB Position vector of B Position vector of A (i j k ) (i j k ) i j k nd C AC Position vector of C Position vector of A (i j k ) (i j k ) i j k A B i j k Now, AB AC i ( ) j ( 8) k ( ) i j k p nd AB AC () ( ) () 9 Thus, re of ABC 9 AB AC sq units Also, DB Position vector of B Position vector of D (i j k ) (i j k ) i j k nd DC Position vector of C Position vector of D (i j k ) (i j k ) i j k i j k Now, DB DC i ( ) j ( ) k ( ) i j k p nd DB DC () ( ) () 9 9 DB DC sq units Hence, re of qudrilterl ABCD re of ABC re of DBC 9 sq units Thus, re of DBC Let the given points e A(k,, ), B(,, ) nd C(,, ) Then, AB Position vector of B Position vector of A nd (i j k ) (k i j k ) ( k)i 9j k BC Position vector of C Position vector of B (i j k ) (i j k ) i j k Given tht the points A, B nd C re colliner Vectors AB nd BC re prllel ~ AB BC i k j k 9 ~
Solution to Prctice Pper Ji B Ki i ( ) j ( ) k ( k 8) i j k i j ( k )k i j k On compring oth sides, we get k r Let I sin r Put sin sin k ( ) sin ( sin ) sin cos sin tn tn sec I tn sec {z} {z } I II tn tn () tn tn tn (sec ) tn tn C r r r tn C tn r r tn tn C mn Given m n ( ) Differentiting oth sides wrt, we get d n d d mn m ( ) n (m ) ( ) mn d m n n n mm (m n) ( ) ( ) mn n m n m m n (m n) ( ) ( ) m n m (m n) n m n ( ) () [Using ()]
Supplement to An Aid to Mthemtics Clss XII n m (m n) ( ) Ji B Ki n m (m n) (m n) ( ) ( ) n (m n) (m n) m ( ) ( ) (n n m n) (m n m m) ( ) ( ) (n m) (n m) mn Second Method: Given m n ( ) Tking log on oth sides, we get mn log (m n ) log ( ) log m log n log ( ) mn m log n log (m n) log ( ) Differentiting oth sides wrt, we get d d d m (log ) n (log ) (m n) [log ( )] m n (m n) d ( ) ( ) (m n) m n ( ) Proceeding s in the first method, we get LHS c c c c c c ( ) c c c c c c c c c c c ( ) c c c c c(c ) c On multipling R, R nd R c On tking common from C, from C nd c from C
Solution to Prctice Pper c c c c ( c ) c c ( c ) c Ji B Ki c c c On ppling R R R R On tking common ( c ) from R On ppling C C C C C C h i On epnding long R ( c ) [{ } ] c RHS Second Method: LHS c c c c c c c c c c c c On tking common from R, from R nd c from R c On ppling R R R R R R On multipling C, C nd C c c c c ( c ) c ( c ) [{ } ] c RHS On ppling C C C C On tking common ( c ) from C h On epnding long C i
Ji B Ki Supplement to An Aid to Mthemtics Clss XII 7 Given differentil eqution is ( )( ) ( ) ( ) ( ) ( ) dv in (), we get dv v v dv v v v dv v v dv v Integrting oth sides, we get v dv v dv v Putting v nd v log v C log C log C, which is the required solution 8 Let B, B nd E e the events defined s follows: B : Letter (ie, messge) hs come from CALCUTTA B : Letter (ie, messge) hs come from TATANAGAR nd E : Two consecutive letters TA re visile on envelope 7 pirs of consecutive letters re Then, P (B ), P (E B ), 7 CA, AL, LC, CU, UT, TT, TA 8 pirs of consecutive letters re P (E B ) P (B ), 8 TA, AT, TA, AN, NA, AG, GA, AR B Bes Theorem, we hve P (B )P (E B ) 7 8 P (B E) P (B )P (E B ) P (B )P (E B ) 7 8 7 Hence, required proilit ()
Solution to Prctice Pper Ji B Ki7 Let B, B nd E e the events defined s follows: B : Person hs lood group O, B : Person hs lood group other thn O nd E : Left-hnded person is selected, 7 P (B ), B Bes Theorem, we hve Then, P (B ), P (E B ) P (E B ) P (B )P (E B ) P (B )P (E B ) P (B )P (E B ) 9 7 9 Hence, required proilit P (B E) 9 Let α e the ngle etween the given lines Eqution of first line is z 7 z ie, 7 Direction rtios of first line re, 7, Eqution of second line is 8 z z ie, Direction rtios of second line re,, Let, 7, c nd,, c Then, cos α p p c c p c c () ()( ) (7)() ( )() p (7) ( ) ( ) () () So, α 9 Hence, the given two lines re perpendiculr Let R denote the rdius, V denote the volume nd A denote the surfce re of sphericl ll t instnt t dv dr A;? Here, dt dt
Supplement to An Aid to Mthemtics Clss XII 8Ji B Ki Now, dv A dt dv k A dt dv kr dt V Also, where k is positive constnt () R dv dr R dt dt dr dv dt R dt () From () nd (), we get dr dt R ( kr ) k Hence, the rdius is decresing t constnt rte Let denote the distnce etween the foot of ldder nd the wll, denote the distnce etween the top of ldder nd the ground nd θ denote the ngle etween the floor nd the ldder t instnt t Here, dθ cm/s m/s;? dt dt Wll Differentiting oth sides wrt t, we get dθ cos θ dt dt sin θ Now, m θ Ground dθ dt cos θ dt dθ dt dθ dt dθ When, we hve rdin/s dt Hence, the required rte of decrese rdin/s Suppose tht Crossing hurdle is considered s tril Let success nd filure for ech tril e defined s follows: Success : Knocking down hurdle nd Filure : Not knocking down hurdle Since, the trils re independent Proilit of filure in ech tril, q nd proilit of success in ech tril, p q Numer of trils, n Let X e rndom vrile defined s the numer of successes
Solution to Prctice Pper Ji B Ki9 We hve, P (X ) n C p q n Required proilit P ( hurdles re knocked down) P ( hurdle is knocked down) P (X ) P (X ) 9 C C 9 9 The life skills the pler must develop to improve his performnce re hrd work, regulrit, determintion, commitment nd sincerit I () Let cos sin ( ) I cos ( ) sin ( ) ( ) I () cos sin On dding () nd (), we get ( ) I cos sin cos sin ( ) I cos sin cos sin Assume f () cos sin f () Then, f ( ) cos ( ) sin ( ) cos sin / I cos sin / On dividing numertor nd sec tn denomintor cos tn Put sec I If, then If, then tn
Supplement to An Aid to Mthemtics Clss XII Ji B Ki ( ) if if ( ) k if if if k if Given f () k Continuit t We hve, f () k lim [f ()] lim B putting lim h h h lim [f ()] lim lim h h So, f is continuous t if lim [f ()] lim [f ()] f () k k ie ie Given A Then, A ( ) ( ) A is non-singulr nd hence invertile Let Aij denote cofctor of ij in A [ij ], then A ( ), A ( ), A ( ), A ( ) A ( ) A ( ), A ( ), A ( ),,, B putting h
Solution to Prctice Pper Ji B Ki A A A dj A A A A A A A nd A (dj A) A Now, given sstem of equtions is A ( ) z 8 z 7 which cn e written in mtri form s 8 7 z AT X B, where AT, X, B 8 z 7 ie, Now, AT A So, the given sstem of equtions is consistent nd independent, ie, it hs unique solution given T X (AT ) B A B T T X A B 8 8 z 7 7,, z Hence, the required solution is,, z Given curves re (i) p ie, ± p To find points of intersection From (i) nd (ii), we hve p ± p p p p 8p (ii) p ie, p
Supplement to An Aid to Mthemtics Clss XII Ji B Ki, p Curves (i) nd (ii) intersect ech other t (, ) nd (p, p) To plot the required region ± p p p p p p p p p p The required region is shown shded in the grph p p Scle Along -is: cm p units p Along -is: cm p units p p p p p p p p p p To find the re of required region p p p p / 8p 8p A p ( ) p p Hence, required re p p sq units Given R Set of ll rel numers nd R {(, ) :, R nd is n irrtionl numer} (i) Refleive: Since, is n irrtionl numer, is true, where R (, ) R So, R is refleive (ii) Smmetric: Let (, ) R, where, R is n irrtionl numer is not n irrtionl numer, for some, R For,, we hve (, ) R ut (, ) / R So, R is not smmetric
Solution to Prctice Pper Ji B Ki (iii) Trnsitive: Let (, ) R nd (, z) R, where,, z R is n irrtionl numer nd z is n irrtionl numer z is not n irrtionl numer, for some,, z R For,, z, we hve (, ) R, (, z) R ut (, z) / R So, R is not trnsitive Given sin () sin sin sin () h i sin sin () i h sin sin () cos sin () Putting sin α, we get () sin α cos α p sin α ± sin α p () ± () Squring oth sides, we get 8 ± Verifiction, we hve [from ()] For! LHS sin sin RHS, we hve [from ()]! LHS sin sin! sin sin RHS For is the required solution 7 Given eqution of plne is ~r (i j k ) p Hence, ~r ( i j k ) p ie, which is of the form ~r (i j ck ) d with,, c, d p
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Also, given point is (,, ) Let,, z Now, distnce of the point (,, z ) from the given plne is given cz d D c ( )() ()() ( )() p p p p ( ) () ( ) Also, distnce etween (,, ) nd (,, ) is p D ( ) ( ) ( ) Given tht product of these distnces is 8 D D 8 p 8 p 8 p±8 p 7, 9 Hence, required vlues of p re 7 nd 9 Given eqution of plne is z P (7,, ) () which is of the form cz d R with,, c, d Direction rtios of norml to the given plne re,, Plne Direction rtios of line perpendiculr to the given plne Q re,, So, the eqution of line through (7,, ) nd perpendiculr to the given plne is 7 z 7 z Let k Then, k 7, k, z k From () nd (), we hve (k 7) (k ) ( k ) k Putting k, in (), we get,, z 8 The point of intersection of the line nd plne is (,, 8) k ()
Solution to Prctice Pper Ji B Ki Required foot of perpendiculr is R(,, 8) Now, length of perpendiculr Distnce etween (7,, ) nd (,, 8) p ( 7) ( ) (8 ) 89 units Let Q(α, β, γ) e the imge point of the point P (7,, ) in the plne Then, R(,, 8) is the mid-point of P Q 7 α β γ Also, coordintes of mid-point of P Q re,, β γ 7α,, 8 So, α, β, γ Hence, the imge point of P in the given plne is (,, ) 8 Let D denote the distnce etween (α, β) on the curve 7 nd (, 7) p D (α ) (β 7) () Also, point (α, β) lies on the curve 7 β α 7 Putting vlue of β in (), we get p (α ) α p D α α α 9 D () D α α α 9 Putting D, we get α α α 9 Then, Now, d α α dα d dα α α α α (α ) α α Also, α, α d α dα d > dα α B second derivtive test, ( D ) is minimum when α So, D is minimum when α Hence, nerest distnce etween soldier nd helicopter is units Putting vlue of α in (), we get D 9 ±
Supplement to An Aid to Mthemtics Clss XII Ji B Ki Let A denote the re of rectngle of length L nd redth B, inscried in the ellipse hving eqution Then, A (L)(B) A LB () Also, point (L, B) lies on the ellipse L B (L, B) L B O ( B ) p L B Putting vlue of L in (), we get p A B B L () ( B )B B B Putting A, we get B B A d B B db Then, d db Now, B B B B B( B ) B d B db Also, B d db B/ < So, second derivtive test, is mimum when B ( B ) ( B > )
Solution to Prctice Pper Ji B Ki7 Putting vlue of B in (), we get r L Putting vlues of L nd B in (), we get A Hence, the re of the required rectngle is sq units 9 Let the distnce trvelled t speed of km/h km nd the distnce trvelled t speed of 8 km/h km Let totl distnce trvelled km We cn represent the given LPP in the following tulr form: Speed km/h Speed 8 km/h Requirement Distnce (km) Mimise Amount spent ( ) 8 At most Time spent (hours) At most Hence, given LPP is, Mimise suject to the constrints:, 8,, We consider the following equtions: 8, ie, 8 8 The fesile region of LPP is ounded, s shown shded in the grph 8 Scle Along -is: cm units Along -is: cm units A B C O 8 8
8Ji B Ki Supplement to An Aid to Mthemtics Clss XII Corner Points Vlue of ( ) A(, ) 8 B, 7 7 8 7 C(, ) O(, ) 8 is the mimum vlue of t corner points 7 8 8 is the mimum vlue of in the fesile region t, 7 7 7 km, Hence, distnce trvelled t speed of km/h 7 8 8 km nd mimum distnce trvelled km distnce trvelled t speed of 8 km/h 7 7 ooo Since, the fesile region is ounded nd
Prctice Pper Time: Hours M Mrks: Generl Instructions (i) All questions re compulsor (ii) Plese check tht this question pper consists of 9 questions (iii) Questions to in Section A re Ver Short Answer Tpe Questions crring mrk ech (iv) Questions to in Section B re Short Answer I Tpe Questions crring mrks ech (v) Questions to in Section C re Long Answer I Tpe Questions crring mrks ech (vi) Questions to 9 in Section D re Long Answer II Tpe Questions crring mrks ech (vii) Plese write down the seril numer of the question efore ttempting it Section A Question numers to crr mrk ech If ~ nd λ, then find the rnge of λ~ Write the function in the simplest form: tn ; < ( if i j If mtri A [ij ], where ij, find the vlue of A if i j Evlute: tn (sec ) cot (cosec ) Section B Question numers to crr mrks ech Evlute: sin ( ) sin ( ) cos tn cos Prove tht: tn 7 For the curve, if increses t the rte of units per second, then how fst is the slope of the curve chnging when? cos ( ) cos ( ) 9 Assume tht in fmil, ech child is equll likel to e o or girl A fmil with three children is chosen t rndom Find the proilit tht the eldest child is girl given tht the fmil hs t lest one girl 8 Solve the differentil eqution: tn Let the function f : R R e defined f () cos, R Show tht f is neither one-one nor onto Prove tht: (~ ~) ~ ~ (~ ~) sin () tn cos () tn If A, B, then find the sin cot () sin tn () vlue of A B
Supplement to An Aid to Mthemtics Clss XII Ji B Ki Section C Question numers to crr mrks ech sin Evlute: cos Prove, using properties of determinnts: ( ) Epress A s the sum of smmetric nd skew smmetric mtri The mgnitude of the vector product of the vector i j k with unit vector long the sum of vectors i j k nd λi j k is equl to Find vlue of λ For the curve, find ll points t which the tngent psses through the origin 7 If, then prove tht ( ) 8 A coin is ised so tht the hed is times s likel to occur s til If the coin is tossed twice, find the proilit distriution nd epecttion of numer of tils 9 Find the eqution of the plne pssing through the point (,, ) nd mking equl intercepts on the coordinte es A plne meets the coordinte es in A, B, C such tht the centroid of ABC is the point z (α, β, γ) Show tht the eqution of the plne is α β γ Prove tht the reltion R on set N N, defined (, ) R (c, d) d c, for ll (, ), (c, d) N N is n equivlence reltion Let A R {} nd B R {} Consider the function f : A B defined f () Show tht f is one-one nd onto nd hence find f cos cos Evlute: cos There re three coins One is ised coin tht comes up with til % of the times, the second is lso ised coin tht comes up heds 7% of the times nd third is n unised coin One of the three coins is chosen t rndom nd tossed, it showed heds Wht is the proilit tht it ws the unised coin? Is it fir to use ised coin for toss efore the strt of mtch? Find the generl solution of the differentil eqution: p
Prctice Pper Ji B Ki Section D Question numers to 9 crr mrks ech A window is in the form of rectngle surmounted semicircle The totl perimeter of the window is m Find the dimensions of the rectngulr prt of the window to dmit mimum light through it Show tht the rdius of the right circulr clinder of gretest curved surfce re which cn e inscried in given cone is equl to hlf of tht of the cone Find the distnce etween the point P (,, 9) nd the plne determined the points A(,, ), B(,, ) nd C(,, ) Evlute: 7 If A, then find A Using A, solve the sstem of equtions: z, z, z If,, z re different nd z z, then show tht z z 8 A o mnufcturer mkes lrge nd smll oes from lrge piece of crdord The lrge oes require sq m per o while the smll oes require sq m per o The mnufcturer is required to mke t lest three lrge oes nd t most twice s mn smll oes s lrge oes If sq m of crdord is in stock, nd if the profits on the lrge nd smll oes re nd respectivel, how mn of ech should e mde in order to mimise the totl profit? Formulte the ove LPP mthemticll nd then solve it grphicll One kind of cke requires g of flour nd g of ft, nd nother kind of cke requires g of flour nd g of ft Find the mimum numer of ckes which cn e mde from kg of flour nd kg of ft, ssuming tht there is no shortge of the other ingredients used in mking the ckes Formulte the ove LPP mthemticll nd then solve it grphicll 9 Find the re ounded the prol nd ooo
Solution to Prctice Pper Given ~ λ Also, λ λ ~ ~ λ~ [Given] [ ~ ] Hence, the required rnge is [, ] Let tn Putting sin α, we get sin α sin α p tn (tn α) α sin tn tn cos α sin α Hence, sin is the required simplest form ( if i j Given A, where ij if i j, Then, So, A,, Hence, A AA tn sec cot cosec h i tn tn cot cot h i n h io tn tn cot cot h i h i sin ( ) sin ( ) sin [( ) ( )] sin ( ) sin ( ) sin ( ) sin ( ) cos ( ) cos ( ) sin ( ) cosec ( ) sin ( ) sin ( ) cosec ( ) [cot ( ) cot ( )] cosec ( ) [log sin ( ) log sin ( ) ] C
Solution to Prctice Pper LHS tn tn cos cos Putting cos α, we get h i h i LHS tn α tn α tn α tn α tn α tn α ( tn α) ( tn α) ( tn α)( tn α) Ji B Ki ( tn α) tn α cos α α cos α cos RHS 7 Let denote the ordinte, denote the sciss nd z denote the slope of the curve t instnt t dz Here, units/s ;? dt dt Now, dz () dt dt dz When, we hve 7 units/s dt Hence, the required rte of chnge 7 units/s z 8 Given differentil eqution is tn cos ( ) cos ( ) ( ) ( ) ( ) ( ) tn cos cos tn cos cos sec tn cos Integrting oth sides, we get sec tn which is the required solution cos sec sin C,
Ji B Ki Supplement to An Aid to Mthemtics Clss XII 9 Totl numer of children in fmil Smple spce, S {, g, g, gg, g, gg, gg, ggg}, where denotes o nd g denotes girl Let E nd F e the events defined s follows: E : The eldest child is girl nd F : The fmil hs t lest one girl Then, E {g, gg, gg, ggg}, F {g, g, gg, g, gg, gg, ggg} nd E F {g, gg, gg, ggg} 7 Also, P (F ), P (E F ) 8 8 P (E F ) 8 Hence, required proilit P (E F ) 7 P (F ) 7 8 Given f : R R defined f () cos (i) One-one: Let, R e n two elements Then, f ( ) f ( ) cos cos Since, f () f (), ut So, f is not one-one (ii) Onto: Let R e n element Then, f () cos For R, there is no R So, f is not onto Hence, f is neither one-one nor onto Let θ e the ngle etween ~ nd ~ We hve, (~ ~) ~ ~ ~ ~ sin θ ~ ~ sin θ ~ ~ ( cos θ) ~ ~ ~ ~ cos θ ~ ~ ~ ~ cos θ ~ ~ (~ ~) sin () tn cos () tn Given A nd B cot () tn () sin sin sin () tn cos () tn Then, A B sin cot () sin tn ()
Solution to Prctice Pper Ji B Ki sin () tn cos () tn sin sin cot () tn () sin () cos () cot () tn () I I Let ( ) sin ( ) cos ( ) ( ) sin cos I () I sin cos () On dding () nd (), we get sin cos I Put cos sin I ( ) sin cos ( ) sin cos I I sin cos If, then If, then h i tn LHS ( )( ) ( )( ) ( ) On ppling R R R R R R
Supplement to An Aid to Mthemtics Clss XII Ji B Ki ( ) ( ) [{( ) ( )} ] On tking common ( ) from R nd R h i On epnding long C ( ) RHS Given A, then AT (A AT )! Since, P T P P is smmetric mtri Let P (A AT )! Since, QT Q Q is skew smmetric mtri Let Q Also, P Q A Hence, A P Q, where P is smmetric mtri nd Q is skew smmetric mtri Let ~ i j k, ~ i j k nd ~c λi j k Then, ~ ~c (i j k ) (λi j k ) ( λ)i j k Unit vector long the sum of vectors ~ ~c is given ~ ~c ( λ)i j k ( λ)i j k r p ~ λ λ ( λ) () ( ) ~c i Now, ~ r λ λ λ λ λ j λ λ i j k λ k λ λ
Solution to Prctice Pper Ji B Ki7 h i i ( ) j { ( λ)} k { ( λ)} λ λ 8i ( λ)j ( λ)k λ λ Given tht the mgnitude of the vector product of ~ with unit vector long ~ ~c is equl to ~ r 8i ( λ)j ( λ)k λ λ 8i ( λ)j ( λ)k λ λ p p ( 8) ( λ) ( λ) λ λ ( λ 8λ) ( λ 8λ) (λ λ ) λ 9 λ 8λ 88 8 8λ λ Given curve is () Let the required point on the given curve e (α, β) β α α () Differentiting oth sides of () wrt, we get α α (α,β) Now, eqution of tngent t (α, β) is β ( α) (α,β) β α α ( α) ie, () Given tht tngent psses through the point (, ) Putting nd in (), we get β α α ( α) β α α From () nd (), we get α α α α 8α 8α 8α (α ) α,, ()
8Ji B Ki Supplement to An Aid to Mthemtics Clss XII Putting α in (), we get β Putting α in (), we get β Putting α in (), we get β (, ), (, ) nd (, ) re the points on the curve Also, ll the three points stisf () Hence, required points re (, ), (, ) nd (, ) p 7 Given p ( ) ( ) ( ) ( ) ( ) ( ) Differentiting oth sides wrt, we get d d ( ) () () ( ) ( ) () ( ) ( ) ( ) 8 Numer of times coin is tossed Suppose tht getting til is considered s success Given tht the hed is times s likel to occur s til ie, P ( hed) P ( til) P ( til) P ( til) P ( til) nd P ( hed) Let X e rndom vrile defined s the numer of successes Then, X cn ttin the vlues,, Now, P (X ) P (oth re heds) 9 P (X ) P ( hed nd til) So, P ( til) P (first is hed nd second is til) P (X ) P (oth re tils) [Fi Cse] [Unfi Cse]!!!!!! [Fi Cse]
Solution to Prctice Pper Ji B Ki9 So, proilit distriution of X is given X P (X) 9 Clcultion of Epecttion X P (X) 9 X P (X) P X P (X) 8 9 Given, plne which mkes equl intercepts (s, ) on the coordinte es Then, eqution of plne, hving intercepts, nd on the coordinte es, is z Hence, epecttion E(X) P X P (X) z () Also, given tht the plne psses through the point (,, ) Putting in (), we get z, which is the required eqution Eqution of plne, hving intercepts,, c on the coordinte es, is z c Given tht plne meets the coordinte es in A, B, C The coordintes of A, B, C re (,, ), (,, ), (,, c) respectivel c Then, the centroid of the ABC is,, Given tht centroid of the ABC is (α, β, γ) c α, β, γ ie, α, β, c γ Putting vlues of,, c in (), we get the eqution of the required plne s z α β γ ()
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Given N Set of ll nturl numers nd R {((, ), (c, d)) : d c; (, ), (c, d) N N} {((, ), (c, d)) : c d; (, ), (c, d) N N} (i) Refleive: Since,, is true, where (, ) N N ((, ), (, )) R So, R is refleive (ii) Smmetric: Let ((, ), (c, d)) R, where (, ), (c, d) N N c d c d ((c, d), (, )) R So, R is smmetric (iii) Trnsitive: Let ((, ), (c, d)) R nd ((c, d), (e, f )) R, where (, ), (c, d), (e, f ) N N c d nd c d e f e f ((, ), (e, f )) R So, R is trnsitive Hence, R is n equivlence reltion (i) One-one: Let, R {} e n two elements Given f : R {} R {} defined f () f ( ) f ( ) Then, So, f is one-one (ii) Onto: Let R {} e n element f () Then, ( ) For ever R {}, we hve So, f is onto Thus, f is ijection nd hence invertile R {} ()
Solution to Prctice Pper So, f : R {} R {} eists nd from (), we hve f () Ji B Ki [ f () f ()] 9 cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos sin [cos cos ] sin C Hence, f is given f () Let B, B, B nd E e the events defined s follows: B : First ised coin is tossed, B : Second ised coin is tossed, B : Unised coin is tossed, P (E B ) P (B ), P (E B ) P (B ), P (E B ) B Bes Theorem, we hve Then, P (B ) nd E : Coin shows hed, 7, P (B )P (E B ) P (B )P (E B ) P (B )P (E B ) P (B )P (E B ) 7 Hence, required proilit No, it is not fir to use ised coin for toss efore the strt of mtch P (B E)
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Given differentil eqution is p p p r dv Putting v nd v in (), we get p dv v v v p dv v dv v Integrting oth sides, we get dv v p log v v log C r log C, log () which is the required solution We know tht window with lrgest re will dmit mimum light Let A denote the re of the window consisting of rectngle of length nd redth surmounted semicircle of dimeter () A Now, perimeter of window, P ( ) () Putting vlue of in (), we get A [ ( )] ( ) da Then, ( ) da Now, ( )
Solution to Prctice Pper d A ( ) Also, Ji B Ki d A ( ) < /() B second derivtive test, A is mimum when Putting vlue of in (), we get ( ) Hence, length nd redth of rectngulr prt re m nd m respectivel Let A denote the curved surfce re of clinder of rdius R nd height H, inscried in given cone of height h, rdius r nd semi-verticl ngle α A RH () A Also, in right ngled ABC, we hve BC tn α AB R tn α h H α B h H R cot α C h H H h R cot α D Putting vlue of H in (), we get R E r A R(h R cot α) A Rh R cot α da (h R cot α) dr da Now, dr () Then, (h R cot α) h R cot α R d A ( cot α) cot α dr Also, h tn α d A dr cot α < [ < α < Rh tn α/ So, second derivtive test, A is mimum when R h tn α ]
Ji B Ki Supplement to An Aid to Mthemtics Clss XII Also, in right ngled ADE, we hve DE tn α AD r tn α h r h tn α h tn α r Hence, required rdius of the right circulr clinder is equl to hlf of the rdius of the cone Thus, R Second Method: Let A denote the curved surfce re of clinder of rdius R nd height H, inscried in given cone of height h nd rdius r A A RH () Also, ABC ADE AB AD h H h BC DE R r B H h hr da h dr r da Now, dr h H hr r Putting vlue of H in (), we get hr hr A R h hr r r C D R E r Then, R r r R d A h dr r Also, d A dr hr r R h r h Rr/ h < r [ h ] [ r, h > ] r Hence, required rdius of the right circulr clinder is equl to hlf of the rdius of the cone So, second derivtive test, A is mimum when R
Solution to Prctice Pper Ji B Ki Given points on plne re A(,, ), B(,, ) nd C(,, ) Let,, z ;,, z ;,, z Then, eqution of plne through A, B nd C is z z z z z z z ( )( ) ( )(8 8) (z )( ) z 7 z 9 z 9 which is of the form cz d with,, c, d 9 Also, given point is P (,, 9) Let,, z 9 Now, distnce of the point (,, z ) from the given plne is cz d D c ()() ( )() ()(9) 9 p () ( ) () Hence, required distnce units Let I I I Now, I Assume f () ( ) Then, f ( ) f () ( ) I ()
Supplement to An Aid to Mthemtics Clss XII Ji B Ki Assume f () f () Then, f ( ) ( ) I ( ) Now, I [log ] [log ] log Putting vlues of I nd I in (), we get I log log 7 Given A Then, A ( ) ( ) ( ) A is non-singulr nd hence invertile Let Aij denote cofctor of ij in A [ij ], then A ( ), A ( ), A ( ), A ( ), A ( ) 9, A ( ), A ( ), A ( ), A ( )
Solution to Prctice Pper A A A A 9 A A nd A (dj A) 9 9 A Now, given sstem of equtions is z z z Ji B Ki7 A dj A A A which cn e written in mtri form s z AX B, where A, X, B z ie, Since, A Given sstem of equtions is consistent nd hs unique solution given X A B X A B 9 z,, z Hence, the required solution is,, z Let z z z z z z z z z z z z z z z z z On tking common from R, from R nd z from R in second determinnt z On ppling C C in second determinnt